Predicting Phenotypic Diversity and the UnderlyingQuantitative Molecular TransitionsClaudiu A. Giurumescu1, Paul W. Sternberg2, Anand R. Asthagiri1*
1 Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California, United States of America, 2 Division of Biology, California
Institute of Technology, Pasadena, California, United States of America
Abstract
During development, signaling networks control the formation of multicellular patterns. To what extent quantitativefluctuations in these complex networks may affect multicellular phenotype remains unclear. Here, we describe acomputational approach to predict and analyze the phenotypic diversity that is accessible to a developmental signalingnetwork. Applying this framework to vulval development in C. elegans, we demonstrate that quantitative changes in theregulatory network can render ,500 multicellular phenotypes. This phenotypic capacity is an order-of-magnitude belowthe theoretical upper limit for this system but yet is large enough to demonstrate that the system is not restricted to a selectfew outcomes. Using metrics to gauge the robustness of these phenotypes to parameter perturbations, we identify a selectsubset of novel phenotypes that are the most promising for experimental validation. In addition, our model calculationsprovide a layout of these phenotypes in network parameter space. Analyzing this landscape of multicellular phenotypesyielded two significant insights. First, we show that experimentally well-established mutant phenotypes may be renderedusing non-canonical network perturbations. Second, we show that the predicted multicellular patterns include not onlythose observed in C. elegans, but also those occurring exclusively in other species of the Caenorhabditis genus. This resultdemonstrates that quantitative diversification of a common regulatory network is indeed demonstrably sufficient togenerate the phenotypic differences observed across three major species within the Caenorhabditis genus. Using ourcomputational framework, we systematically identify the quantitative changes that may have occurred in the regulatorynetwork during the evolution of these species. Our model predictions show that significant phenotypic diversity may besampled through quantitative variations in the regulatory network without overhauling the core network architecture.Furthermore, by comparing the predicted landscape of phenotypes to multicellular patterns that have been experimentallyobserved across multiple species, we systematically trace the quantitative regulatory changes that may have occurredduring the evolution of the Caenorhabditis genus.
Citation: Giurumescu CA, Sternberg PW, Asthagiri AR (2009) Predicting Phenotypic Diversity and the Underlying Quantitative Molecular Transitions. PLoSComput Biol 5(4): e1000354. doi:10.1371/journal.pcbi.1000354
Editor: Christopher Rao, University of Illinois at Urbana-Champaign, United States of America
Received December 29, 2008; Accepted March 10, 2009; Published April 10, 2009
Copyright: � 2009 Giurumescu et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Institute for Collaborative Biotechnologies Grant DAAD 19-03-D-0004 from the U.S. Army Research Office (to A.R.A.),the Center for Biological Circuit Design at Caltech, and the Jacobs Institute for Molecular Engineering for Medicine. P.W.S. is an investigator with the HowardHughes Medical Institute. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: [email protected]
Introduction
During development, regulatory signaling networks instruct cell
populations to form multicellular patterns and structures. To what
extent perturbations in the quantitative performance of these
networks may lead to phenotypic changes remains unclear.
Experimental genetics studies typically uncover mutant pheno-
types that emerge from extreme modes of perturbation (e.g.,
knockout or overexpression) [1,2]. However, there is ample
evidence that biological networks operate amidst quantitative
fluctuations [3–6]. The sources of these quantitative perturbations
include stochastic behavior, population heterogeneity, epigenetic
effects and environmental changes.
The fundamental question then is how much phenotypic
variation is possible by quantitative perturbations in network
performance without wholesale changes to network topology. On
the one hand, we may expect that the wild-type multicellular
phenotype may be highly robust to quantitative variations. Indeed,
computational analysis of the Drosophila segment polarity network
demonstrated the robustness of the wild-type multicellular pattern
to significant parameter changes [7]. This robustness may be a
more pervasive property of developmental regulatory networks
that allows their modular utilization in different multicellular
geometries and developmental contexts [8]. On the other hand,
for a given multicellular system, some degree of fragility in the
regulatory network is essential for evolutionary diversification.
New multicellular phenotypes must be accessible through
modifications to the underlying regulatory network, providing
avenues for sampling new phenotypes that may be more beneficial
under different selective pressures.
The extent to which this phenotypic diversification must involve
a topological overhaul of the regulatory network as opposed to
quantitative changes to a fixed network topology remains unclear.
Closely related species may have evolved by subtle, quantitative
changes in network interactions rather than large-scale changes to
network topology. Indeed, there is evidence for such ‘‘quantitative
diversification’’ of phenotypes in the evolution of maize and finch
beaks [9,10]. However, analyzing extant species identifies only
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quantitative changes that have withstood selection and conceals
the complete phenotypic diversity that a regulatory network can
render. Meanwhile, experimentally reconstructing that diversity
faces the challenge of systematically imposing quantitative
regulatory perturbations in vivo and scoring the numerous
phenotypes that would be generated.
Computational modeling has proven to be a useful tool for
predicting multicellular patterns and morphology based on the
underlying regulatory mechanisms [7,11–18]. Thus, such models
may provide an effective framework to explore the full diversity of
phenotypes that is accessible through quantitative changes to a
particular developmental regulatory network. Here, we develop a
computational approach to analyze quantitatively the phenotypic
diversity of C. elegans vulval development. The C. elegans vulva
develops from an array of six precursor cells that commit to a
spatial pattern of distinct fates (Figure 1) [19,20]. We have
described previously a mathematical model of the regulatory
network that controls C. elegans vulval development and elucidated
potential quantitative advantages of the biochemical coupling in
this signaling network [21]. In this work, we extend this
mathematical model of the signaling network to make predictions
about the range of phenotypes that this network can render. We
probed whether this developmental network is so robust to
parameter changes that only a narrow set of multicellular
phenotypes is possible. Or, can quantitative variations give rise
to a broader range of phenotypes? In contrast to other recent
models of C. elegans vulval development [12,13], our model
incorporates directly the underlying molecular mechanisms and
the quantitative strength of these molecular regulatory pathways.
Thus, it provides the necessary foundation for examining
quantitative diversification of multicellular phenotype.
Our computational analysis reveals that a significant amount of
phenotypic diversity is achievable through quantitative changes to
the regulatory network. Thus, this developmental regulatory
network is not ‘‘wired’’ to generate robustly only the wild-type
phenotype. Furthermore, the phenotypes predicted by the model
include not only those observed in C. elegans, but also those found
exclusively in several closely-related species [22]. Thus, our model
predictions validate the hypothesis that quantitative changes to a
common regulatory network have occurred during the diversifi-
cation of several species within the Caenorhabditis genus. Further-
more, by applying our modeling framework to analyze published
experimental phenotypic data, we extract the quantitative
regulatory differences that may have accrued during the evolution
of three major species of the Caenorhabditis genus.
Results and Discussion
Gauging the phenotypic capacity of the vulvaldevelopmental network
We sought to better understand how much phenotypic diversity
a developmental regulatory network can produce through
quantitative changes without altering the network architecture.
To conduct this analysis, we started with our previously reported
mathematical model of the regulatory network that controls vulval
development in C. elegans [21]. This model uses ordinary
differential equations to track the activity of two key signals in
each precursor cell: MAP kinase and the lateral Notch signal
(details are provided in Materials and Methods). The levels of
these two signals are then used to predict the fate of each cell. The
Author Summary
The diversity of metazoan life forms that we experiencetoday arose as multicellular systems continually samplednew phenotypes that withstood ever changing selectivepressures. This phenotypic diversification is driven byvariations in the underlying regulatory network thatinstructs cells to form multicellular patterns and structures.Here, we computationally construct the phenotypicdiversity that may be accessible through quantitativetuning of the regulatory network that drives multicellularpatterning during C. elegans vulval development. We showthat significant phenotypic diversity may be sampledthrough quantitative variations without overhauling thecore regulatory network architecture. Furthermore, bycomparing the predicted landscape of phenotypes tomulticellular patterns that have been experimentallyobserved across multiple species, we systematicallydeduce the quantitative molecular changes that may havetranspired during the evolution of the Caenorhabditisgenus.
Figure 1. Wild-type patterning of C. elegans vulva. The anchor cell (AC) stimulates the vulva precursor cells Pn.p with LIN-3 in a graded manner.These cells laterally interact with their neighbors through the LIN-12 pathway. The crosstalk between LIN-3 and LIN-12 signaling results in the wild-type pattern of differentiation 3u3u2u1u2u3u. In the wild-type organism, the 1u vulval lineage generates progeny that forms the orifice and connects tothe uterus, while the 2u vulval lineage generates progeny that form the vulval lips and connect to the body epidermis. The daughters of the 3u cellsfuse to the surrounding syncytium and do not contribute to the vulval tissue.doi:10.1371/journal.pcbi.1000354.g001
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model consists of eight dimensionless parameters whose values
influence the pattern of fate choices (Figure 2A). To determine the
phenotypes that are accessible through quantitative modulation of
the network, we allowed each parameter to vary across a broad
range of physiological values (Materials and Methods). For each
combination of parameter values, the multicellular phenotype was
computed. In this manner, the multidimensional parameter space
was divided into sub-regions associated with specific multicellular
phenotypes (Figure 2B).
This phase diagram of phenotypes represents the predicted
multicellular patterns that the vulval developmental network can
produce. Extreme values along each parameter axis emulate the
classical experimental scenario where specific molecular pathways
are eliminated (e.g., knock out) or overexpressed. Away from these
extremes, the phase diagram represents phenotypes that are
predicted to occur when regulatory mechanisms are tuned
quantitatively without wholesale changes to network topology.
Thus, by counting the number of unique phenotypes that exist in
this multidimensional parameter space, we sought to quantify the
‘‘phenotypic capacity’’ of the C. elegans vulval signaling network.
Our calculations show that the phenotypic capacity has an
upper limit. That is, even as the parameter space is broadened, the
number of distinct phenotypes saturates at approximately 560
multicellular patterns (Figure 2C). This result reveals that the
developmental network is not constrained to a few outcomes. The
wild type and a handful of well-studied mutant phenotypes by no
means represent the phenotypic capacity of this system. Further-
more, in this six-cell system there are four fates possible to each cell
(see Materials and Methods). Hence, the theoretical upper limit to
the number of phenotypes is 4,096. Our model predicts that the
molecular network constrains the system from accessing ,85% of
the theoretically possible phenotypes.
To better understand how the phenotypes are represented in
parameter space, we determined the amount of parameter space
associated with each phenotype (see Materials and Methods).
Phenotypes that occur only at a few points in parameter space may
be inaccessible experimentally, while their counterparts occupying
a large fraction of parameter space may represent the more
tangible outcomes. The distribution of Parameter Space Occu-
pancy (PSO) resembles a log-normal distribution (m= 219.60,
s= 4.90) with a slight positive skew (Figure 2D). On the low end of
the distribution, our model predicts 19 phenotypes that are two
standard deviations below the mean PSO (Table S4), and 9 of
these phenotypes do not entail the mixed ‘m’ cell fate (Table S1).
Consistent with this prediction, none of these predicted pheno-
types are among the well-studied experimentally observed
Figure 2. The predicted phenotypic diversity accessible to the vulval developmental network. (A) Model parameters. The model haseight dimensionless parameters associated with the various molecular interactions known to contribute to the specification of vulval precursor cells(see also Materials and Methods). (B) Schematic of the phenotype phase diagram. This diagram portrays a simplified, three-dimensional version of the8-dimensional phenotype phase diagram. Each axis represents a model parameter. Each point in parameter space yields a specific multicellularphenotype, such as the wild-type (3u3u2u1u2u3u, black). (C) The total number of predicted phenotypes eventually saturates as the volume of theparameter space is expanded. (D) Distribution of parameter space occupancy (PSO). A histogram depicting the number of phenotypes (bars)occupying different fractions of the parameter space. This histogram is compared to a log-normal distribution (filled circles). The arrows indicate thePSO values of some experimentally observed phenotypes.doi:10.1371/journal.pcbi.1000354.g002
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phenotypes. These highly unlikely outcomes reduce our evaluation
of the overall phenotypic capacity of this system.
Meanwhile, on the other end of the distribution, a small subset
of phenotypes occupies a disproportionately large portion of the
parameter space (Figure 2D). Within the positive skew is the wild-
type phenotype, consistent with a previous study that showed that
the developmental segment polarity network robustly produces the
wild-type multicellular pattern [7]. Extending beyond the wild-
type phenotype, our model predicts an additional 33 phenotypes
with PSO values that are two standard deviations above the mean
(see Table S5 for a list of these phenotypes), 25 of which do not
entail the mixed ‘m’ cell fate. These phenotypes are highly
represented in parameter space and suggest that significant
phenotypic diversity may be sampled by tuning quantitatively a
common underlying regulatory network. In fact, consistent with
model predictions, several of these 25 phenotypes have been
observed in C. elegans genetics experiments [23,24]. However, 10 of
these 25 phenotypes have not been reported and are novel
predicted phenotypes for future experimental validation.
To further evaluate these 10 novel phenotypes, we developed
two metrics that provide additional insights into how phenotypes
are distributed in parameter space. While the PSO metric
quantifies what fraction of points in parameter space are
associated with a particular phenotype, it does not report how
these points are distributed in parameter space. One extreme is
that the parameter points associated with a phenotype are
disjointed and scattered throughout parameter space (Figure 3A).
In this case, a perturbation in any parameter value would alter the
phenotype, i.e. the phenotype would be highly fragile to parameter
changes. The other extreme is that the parameter points are
contiguous and clustered together into a subspace. In this scenario,
the phenotype would be more robust to parameter variations.
However, the level of robustness would depend on the shape of the
phenotype subspace. A phenotype subspace that contains a high
Figure 3. Robustness of phenotypes that are highly represented in parameter space. (A) Schematic of the Connectivity and Shape (CS)metric of phenotype robustness to parameter changes. The connectivity and shape metric measures the overall likelihood of staying in the currentphenotype upon effecting a random unit-value parameter change from a randomly selected point in the phenotype. Isolated points in the parameterspace do not contribute to the CS metric as unit-value parameter changes starting at such points would lead to exiting the phenotype. Interior pointshowever are fully connected to neighbors and contribute the most to the CS metric. (B) Schematic of the Mean Path Length (MPL) metric ofphenotype robustness to parameter changes. The mean path length metric measures the average number of unit-value parameter changes to exitthe current phenotype. The larger the value, the more robust the phenotype to random parameter changes. (C) The CS and MPL of 26 phenotypeswith the greatest PSO and 2 other experimentally observed phenotypes (2u1u2u1u2u1u and 1u2u2u1u2u1u). Experimentally observed phenotypes aredenoted by blue symbols, while novel phenotypes that are not observed in C. elegans are denoted by green symbols. Dotted lines identify the valueof CS and MPL for the wild-type phenotype. While MPL and CS follow each other monotonically, the CS is a better metric of the phenotyperobustness at CS/MPL values lower than 0.5/1.0, while the MPL is a better metric at values higher than these thresholds.doi:10.1371/journal.pcbi.1000354.g003
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fraction of points at the ‘‘surface’’ (i.e., borders parameter points
belonging to another phenotype) would be less robust than a
phenotype where all its parameter points are tightly packed into a
subspace with minimal exposure to other phenotypes.
To capture these aspects of how parameter points of a particular
phenotype are distributed in parameter space, we developed a
Connectivity and Shape (CS) metric (Materials and Methods). The
value of the CS metric is bounded between 0 and 1 and represents
the average likelihood that for any point in phenotype subspace, a
unit change in any single parameter value maintains the
phenotype (Figure 3A). Thus, a CS value of 0 would refer to a
highly fragile phenotype whose points in parameter space are
‘‘isolated’’ or surrounded by other phenotypes. In contrast, a CS
value of near 1 would refer to a highly robust phenotype for which
most of the points in its parameter subspace are surrounded by
other points associated with the same phenotype. As a comple-
mentary approach to gauge the robustness of a phenotype to
parameter changes, we quantified the Mean Path Length (MPL) as
the average number of unit changes or ‘‘jumps’’ in parameter
values needed to start from any point within a phenotype subspace
and land on a foreign phenotype (Figure 3B, Materials and
Methods)[25]. Large values of MPL indicate that many changes in
parameter values are needed to change phenotype, signifying a
highly robust phenotype.
We calculated the MPL and CS metrics for the 26 phenotypes
with the highest PSO, including the wild-type phenotype
(Figure 3C). In addition, we computed these metrics for two
phenotypes (1u2u2u1u2u1u and 2u1u2u1u2u1u) that occupy less
parameter space (ranked 78th and 79th, respectively, in terms of
PSO, Figure 2D) but are well-established experimental outcomes.
Among these 28 phenotypes, our calculations show a high
correlation between MPL and CS, suggesting that these two
metrics are equivalent ways to gauge the robustness of a phenotype
to parameter variations. The model predicts seven phenotypes
with CS and MPL values greater than that of wild type. All seven
are experimentally observed in C. elegans, suggesting that
robustness, as quantified by these metrics, may be an important
determinant of experimental realizability. Meanwhile, there are 20
phenotypes with CS/MPL metrics lower than the wild type.
Among these 20, ten have been observed in C. elegans genetics
experiments, while the remaining 10 are the aforementioned novel
phenotypes that have not been observed in C. elegans. Notably, the
CS and MPL values of some of these novel phenotypes (e.g.,
2u2u2u3u2u2u, 3u2u2u3u2u2u, and 3u2u3u1u3u2u) falls within the
range of experimentally observed counterparts, suggesting that
these novel phenotypes may be the most realizable experimentally
upon performing the correct manipulations in the LIN-3/MAP
kinase and the LIN-12 pathways.
Identifying optimal molecular perturbations to renderspecific mutant phenotypes
Having predicted novel phenotypes and the experimental
realizability of these outcomes, a key question is how does one
render such phenotypes experimentally? The classical computa-
tional approach is to choose reference parameter values for the
wild-type phenotype and then to test the effect of specific
parameter perturbations. The choice of parameter perturbation
is motivated typically by a corresponding mutation that has been
performed experimentally with the goal of determining whether
the predicted phenotype matches the experimental outcome. The
pitfall, however, is that suitable reference parameter values for the
in vivo biochemistry of signaling pathways in live worms are
unknown. Furthermore, worms are not quantitative clones, and
each worm is likely to differ in its parametric settings. Finally, the
execution of a particular experimental perturbation is unlikely to
be realized in the same quantitative manner in each worm in every
trial.
Based on these considerations, we take a different approach that
is enabled by the phase diagram of phenotypes that we have
computed for this system. Using this phase diagram, we determine
all possible single-parameter changes (i.e., single mutations) that
successfully transition the wild-type phenotype into a mutant
phenotype of interest. The fraction of these successful single-
parameter changes that is associated with a particular parameter
reveals the relative efficacy with which that parameter perturba-
tion ‘‘transitions’’ the wild-type phenotype into the mutant
outcome (Figure 4A and Materials and Methods). In this manner,
these computations yield a transition probability that an increase
(or decrease) in each parameter will shift the phenotype from wild
type to a mutant pattern. Parameter changes with a higher
transition probability have a greater likelihood of generating the
desired mutant phenotype. Thus, this approach is the computa-
tional equivalent of a random genetic screen that evaluates all
possible mutations to determine the most effective ones that lead to
the mutant phenotype of interest.
To test initially this approach, we applied it to mutant
phenotypes that have been well established by genetics experi-
ments in C. elegans. We first predicted the best single-parameter
changes needed to transform the wild-type organism into a
vulvaless mutant. Vulvaless phenotypes have been observed in
genetics experiments and occur when all vulval precursor cells
acquire the 3u fate [2,24,26]. Our model predicts that the best way
to render the 3u3u3u3u3u3u phenotype is by decreasing the level of
inductive signaling (Figure 4B). This prediction is consistent with
experiments in which anchor cell ablation yields the uninduced all-
3u fate pattern [27].
In the other extreme of phenotypes, mutant worms with
multiple vulvae have been observed when the inductive signaling
pathway is hyperactivated [28–30]. In these mutants, the vulval
precursor cells acquire an intriguing alternating pattern of
2u1u2u1u2u1u where each 1u cell produces an invagination [31].
Consistent with this experimental observation, the model predicts
an increase in inductive signal as one of the most prominent ways
to yield this alternating phenotype (Figure 4C).
In addition, because all possible single mutations are evaluated,
our model analysis predicts additional ‘‘equivalent mutations’’ that
would render the same 2u1u2u1u2u1u phenotypic outcome
(Figure 4C). One of these equivalent mutations is to flatten the
gradient in soluble inductive factor (Figure 4C). This particular
prediction is remarkably consistent with what has been recently
uncovered about the most classical experimental mutation to yield
this phenotype. The loss of lin-15 has been shown to cause the
secretion of LIN-3 from the surrounding cells, an event that would
ablate the gradient [32]. A second equivalent mutation predicted
by the model is an increase in the threshold of lateral signaling
needed to inhibit the MAP kinase pathway (kL). This prediction
for generating a well-established phenotype through a non-
canonical perturbation is testable experimentally by decreasing
the binding affinity of the lateral signaling transcription complex
(LAG-1:LIN-12-cyto) to LBS elements in the cis-regulatory regions
of the genes that negatively regulate inductive signaling (ark-1, lip-
1, lst-1,2,3,4) [33]. This mutation would require greater lateral
signaling to inhibit the inductive MAP kinase pathway and would
be an indirect way to inflate the inductive signaling activity,
conceptually consistent with the direct hyperactivation of the
inductive signaling pathway.
An intriguing feature of mutants, such as lin-15(lf) [24,31] and
let-60(gf) [34], is that the observed multicellular pattern is variable.
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In addition to 2u1u2u1u2u1u, the other prominent outcome is
1u2u2u1u2u1u. There are several possible sources of variability [5].
The quantitative levels and interactions of signaling molecules may
differ among wild-type organisms in which the mutation is
performed; thus, their response to a specific perturbation may
produce different outcomes. Alternatively, even if two organisms
were ‘‘quantitative clones,’’ the magnitude of a perturbation being
introduced by the mutation may vary; for example, the amount of
RNAi delivered may be different. Finally, even if the perturbation
and the wild-type organisms were exactly the same, the execution
of the molecular network may deviate due to stochastic effects.
Regardless of the source of variability, the key question we
focused on is why this variability would produce these two
particular outcomes and not others. We hypothesized that in the
parameter space, variable mutant phenotypes may lie in the same
general direction from the wild-type phenotype. That is, because
the starting point, the extent of perturbation and the execution of a
perturbation may differ (Figure 4A), the target points in parameter
space on which these perturbations land will vary but lie within a
common vicinity. To test this hypothesis, we determined what
other phenotypes would be predicted by the model upon
increasing the inductive signal (Figure 5A) or flattening the
gradient (Figure S3). Indeed, the 1u2u2u1u2u1u phenotype is
predicted to occur in response to both perturbations, revealing that
the variable mutant phenotypes lie in the same direction in
parameter space from the wild-type phenotype. Furthermore, our
model predicts that converting the wild-type phenotype to either
the 2u1u2u1u2u1u or 1u2u2u1u2u1u phenotypes would require
approximately the same amount of increase in inductive signal
(Figure 5B). These predictions confirm the hypothesis that these
two phenotypes may co-occur because these outcomes exist at
similar positions relative to the wild-type phenotype in the
multidimensional parameter space.
Model-based testing of the quantitative diversificationhypothesis
An apparent conundrum in our model predictions is that when
inductive signal is increased, the number of predicted phenotypes
is far greater than that observed experimentally in C. elegans
(Figure 5A). In fact, similar calculations show that phenotypes
other than 3u3u3u3u3u3u are possible when the level of inductive
signal is decreased (Figure 5A). Why then is the remarkably rich
set of predicted phenotypes vastly under sampled in experiments
with C. elegans? One possibility is that our model predicts
phenotypes that may occur when the inductive signal is tuned to
intermediate levels; such phenotypes may not be sampled by
Figure 4. Quantitatively predicting the optimal molecular perturbations needed to generate specific mutant phenotypes. (A)Schematic for counting phenotype transitions made possible by single mutations. Subspaces in the 8-dimensional parameter space are occupied bydifferent phenotypes. This diagram portrays a simplified version of the phenotype phase diagram with a single parameter Pk isolated on the x-axisand all other parameters denoted on the y-axis. The transition from iRj can occur by a change in a single parameter Pk, but the transition iRj9cannot. By counting the number of successful single mutations (iRj) for each parameter Pk, we quantify the relative efficacy of each parameter torender a specific phenotype transition (WRM) (see also Materials and Methods). (B,C) The relative probability of inducing a transition from the wild-type phenotype to the 3u3u3u3u3u3u phenotype (B) or the the hyperinduced 2u1u2u1u2u1u phenotype (C) by decreasing (filled columns) or increasing(open columns) the values of parameters indicated on the x-axis. The y-axes report the mean relative transition probability averaged over a broadcombination of threshold values for fate-determining signals, and the error bars denote the standard deviation (see also Materials and Methods). Thesize of the error bar reveals that model predictions are robust to variations in the threshold values of fate-determining signals.doi:10.1371/journal.pcbi.1000354.g004
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classical genetics experiments that typically involve knock-out or
strong overexpression strategies. Another hypothesis is that our
model predicts phenotypes that arise not only in C. elegans, but also
in several closely related species. Several members of the
Caenorhabditis genus undertake a similar step in vulval development
where precursor cells commit to a 3u3u2u1u2u3u wild type pattern
[22,35,36]. Compelling recent experiments have revealed that
tuning the level of inductive signal produces distinct species-
specific mutants even though the starting wild-type phenotype is
the same (Figure 6A) [22]. Since vulval development in all of these
species involves the same regulatory ‘‘parts’’ (EGF and Notch
signaling), these experimental results have raised the intriguing
hypothesis that a common regulatory network has quantitatively
diversified, so that the network still produces the wild-type
phenotype, but when quantitatively perturbed, each species has
access to unique phenotypes (Figure 6B).
Experimentally testing this hypothesis of quantitative diversifi-
cation would involve uncovering the regulatory network driving
vulval development in each member of the Caenorhabditis genus and
proving that the network architecture is indeed the same. This
approach raises practical hurdles of performing numerous genetics
experiments across multiple species. A deeper challenge is that it is
difficult to prove unequivocally that the regulatory network is the
same between two species, since one cannot rule out the existence
of an undiscovered mechanism. On the other hand, a modeling
framework can be particularly effective in testing the quantitative
diversification hypothesis. A model can directly test whether the
proposed vulval regulatory network that has been inferred from
studies in C. elegans is capable of rendering the breadth of
phenotypes observed across multiple species solely through
quantitative changes in regulatory mechanisms.
To conduct this analysis, we compared our predicted pheno-
types (Figure 5A) to the experimentally observed phenotypes in the
three major members of the Caenorhabditis genus, C. elegans, C.
briggsae and C. remanei (Figure 6A). We find that 8 of the 9
experimentally observed phenotypes across the three species are
captured by our model predictions. These results demonstrate
unequivocally that a common vulval developmental network is
capable of producing a significant fraction of the phenotypic
diversity observed in the three major members of the Caenorhabditis
genus. Thus, the model provides new and strong support for
quantitative diversification of a common vulval developmental
network during the evolution of the Caenorhabditis genus.
Where the model fails also provides intriguing insight. Our
model does not predict the 3u2u2u2u3u phenotype that occurs
when inductive signal is decreased moderately in C. briggsae. C.
briggsae is phylogenetically closer to C. remanei than to C. elegans
[22,36], suggesting that quantitative diversification hypothesis may
fail to explain fully how C. briggsae and C. elegans vulval regulatory
networks have diverged. In addition, the model predicts several
phenotypes that are not found in C. elegans, C. briggsae and C. remanei
(Figure 5A). These additional phenotypes may occur in other
members of the Caenorhabditis genus. The seminal dataset collected
by Felix in fact spans eight additional species. We are currently
developing algorithms for systematically clustering and comparing
model-predicted phenotypes to this larger experimental dataset.
Meanwhile, an important feature of the experimental data
gathered by Felix is that species-specific phenotypes emerge only
when the inductive signal is tuned to a certain quantitative level
(Figure 6B). The 3u3u3u3u3u (Class A) and 1u1u1u1u1u (Class D)
phenotypes are observed only when inductive signal is strongly
decreased or increased, respectively; meanwhile Class B
(3u2u3u2u3u, 3u2u2u2u3u and 3u3u1u3u3u) and Class C (1u2u1u2u1u,2u2u1u2u1u, 2u2u1u2u2u and 2u1u1u1u2u) phenotypes occur upon
moderate decrease and increase in inductive signal, respectively.
Thus, a more rigorous test of quantitative diversification is not
only to prove that a common regulatory network can render the
breadth of experimentally-observed phenotypes, but also to
demonstrate that the predicted phenotypes occur only when the
network is tuned in the appropriate quantitative manner. To
Figure 5. The hierarchy of phenotypes predicted to occur through quantitative changes in morphogen level. (A) The relativeprobability (x-axis) of reaching different mutant phenotypes (y-axis) upon decreasing (filled bars) or increasing (open bars) the amount of LIN-3morphogen. The predicted phenotypes denoted with an asterisk are ones observed experimentally in C. elegans. Bold-faced phenotypes correspondto multicellular patterns observed experimentally in the three major species of the Caenorhabditis genus (see Figure 6A). (B) The predicted foldchange in inductive signal (x-axis) necessary to convert the wild-type phenotype into underinduced phenotypes (Fold change ,1) and over-inducedphenotypes (Fold change .1). The phenotypes listed on the y-axis correspond to the bold-faced phenotypes in panel (A). The list has been re-sortedaccording to the required fold change in inductive signal. The shading of bars corresponds to the shading of experimentally observed phenotypes inFigure 6A.doi:10.1371/journal.pcbi.1000354.g005
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undertake this more rigorous test of quantitative diversification, we
determined the amount of change in inductive signal needed to
render the predicted phenotypes. The predicted quantitative
hierarchy of phenotypes (Figure 5B) directly matches experimental
observations (Figure 6B), providing stronger evidence to support
the quantitative diversification hypothesis.
The model predictions directly validate the hypothesis that the
parameter space associated with the wild-type phenotype actually
contains several subspaces, each representing different species. A
key question is which subspace of parameter values corresponds to
each species (Figure 6A). The answer to this question would reveal
how the quantitative settings of this developmental network have
evolved during the emergence of the Caenorhabditis genus. To
address this question, we analyzed more closely the layout of
phenotypes in the multidimensional parameter space. We know
that each species produces different phenotypes when the level of
inductive signal is changed (Figure 6B) [22]. For example, C. elegans
transitions from the phenotype 3u2u1u2u3u (WT) to 3u3u1u3u3uwhen inductive signal is reduced moderately; meanwhile, C.
remanei forms 3u2u3u2u3u upon intermediate reductions in
inductive signal. In both species, a strong reduction in inductive
signal produces 3u3u3u3u3u. Therefore, by identifying the subset of
wild-type parameter values that produce a WTR3u3u1u3u3uR
3u3u3u3u3u transition versus WTR3u2u3u2u3uR3u3u3u3u3u3utransition upon reducing inductive signal, we isolated the C.
elegans and C. remanei parameter subspaces (Materials and
Methods). Similarly, C. briggsae forms patterns with adjacent 1ufates upon mild increase of inductive morphogen signal, while C.
elegans requires a strong increase in morphogen activity to render
such outcomes. Therefore, by distinguishing between WT
R1u2u1u2u1uR1u1u1u1u1u transitions and WTR2u1u1u1u2uR1u1u1u1u1u transitions, we identified the subset of wild-type
parameter values that correspond to C. elegans and C. briggsae
subspaces. We find that C. elegans represents 41.0167.90%, C. briggsae
represents 3.7161.95%, and C. remanei represents 41.3168.20% of
all wild-type space points. The remaining 13.9762.25% of wild-type
space points represents transition patterns that are inconsistent with
experimental results for these three species.
Having identified the sub-region of wild-type parameter space
belonging to C. elegans, C. briggsae and C. remanei, we determined
how the parameters differ among these species (Figures 7A and
7B). The model identifies two potential groups of parameters. The
values of the first group may be higher or lower in C. elegans
relative to C. remanei (DI, x, l, h and kL) or C. briggsae (l, w and kM).
In contrast, the model predicts that a second group of parameters has
changed in a biased manner, either selectively increasing or
Figure 6. (A) Summary of experimentally observed phenotypes in three different species of the Caenorhabditis genus. Distinct phenotypeshave been reported upon increasing or decreasing the level of inductive signal (I) in three major members of the Caenorhabditis genus [22]. The wild-typephenotype (center) is common to all three species. The colored lines denote the species-specific progression of phenotypes as the level of inductive signal(I) is modulated in C. elegans (red), C. briggsae (blue) and C. remanei (green). Phenotypes were not reported in C. remanei upon increasing inductive signalabove the wild-type level [22]; therefore, green arrows are not drawn to the right of the wild-type phenotype. The phenotypes are grouped into Classes A,B, C, and D and are shaded (white, light grey, dark grey and black) according to the amount of perturbation in inductive signal that rendered each mutant.In some species the first pre-cursor cell (P3.p) is not competent to participate in vulval development, and therefore, it is designated as ‘x’. (B) Thequantitative diversification hypothesis. Species belonging to the Caenorhabditis genus all produce a common wild-type phenotype using a commonregulatory network that performs with quantitative differences in parameter settings. Thus, C. elegans (red), C. briggsae (blue) and C. remanei (green) arehypothesized to occupy different subspaces within the wild-type parameter space. This quantitative diversification has been proposed to explain the factthat changes in the level of inductive signal produce species-specific mutant phenotypes.doi:10.1371/journal.pcbi.1000354.g006
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decreasing, during the evolution of C. elegans, C. remanei, and C.
briggsae. The value of w is higher in C. remanei than in C. elegans,
indicating that inductive signaling produces a stronger lateral signal
in C. remanei (Figure 7A). Furthermore, the threshold of inductive
signaling (kM) needed to trigger the lateral signal is lower in C.
remanei. Taken together, these predictions reveal that the ability to
send out lateral signals is far stronger and more sensitive to inductive
signaling in C. remanei than in C. elegans. On the other hand, C. briggsae
deviates from C. elegans primarily in the ability to receive lateral
signals (h) and thereby inhibit inductive signaling (x, kL). Inductive
signaling is predicted to be more sensitive to lateral inhibition in C.
elegans than in C. briggsae (lower x and higher kL in Figure 7B).
These results reveal that during evolution, the members of the
Caenorhabditis genus have taken remarkably divergent paths in
quantitatively modulating a common developmental signaling
network. We demonstrate that the underlying quantitative
molecular changes can in fact be inferred from experimental
observations of phenotypic variability. This inference requires a
computational approach, since the underlying molecular signaling
network is highly interconnected and its relation to emergent
multicellular phenotypes is non-intuitive. Our approach hinges on
a mathematical framework for predicting multicellular phenotypes
from the underlying signaling network and a broadly applicable
computational approach to analyze the phenotypic landscape.
With growing interest in quantitative mechanistic models of
developmental systems [14,16,18], the computational approach
described here will likely find broad application in other
developmental contexts and offers a systematic approach to
mapping the quantitative regulatory changes that have given rise
to divergent developmental phenotypes.
Materials and Methods
Computational model of C. elegans vulval developmentSignaling network and model equations. The vulva in C.
elegans and related species develops from a set of equivalent vulva
precursor cells (VPCs) labeled Pn.p (n = 3 to 8) in Figure 1 [20].
These cells are arranged linearly along the antero-posterior axis of
the body. During the third stage of larval development, the VPCs
receive a spatially graded EGF-like stimulus (LIN-3) from the
anchor cell (AC) in the gonad. Binding of LIN-3 to its receptor
LET-23 activates the MAP kinase MPK-1 and induces their
differentiation. In addition to the soluble LIN-3 signal, juxtracrine
interactions through Notch-like receptor LIN-12 contribute to
VPC differentiation. Together, the inductive LIN-3 signal and the
lateral Notch signal establish a pattern of VPC differentiation
(3u3u2u1u2u3u) in wild-type organisms. Only the VPCs committed
to 1u and 2u fates contribute to vulva formation through cell
divisions and spatial rearrangements of the daughter cells;
meanwhile, the daughters of the 3u-committed VPC fuse to the
hypodermal syncytium.
We previously described a mathematical model of the LIN-3/
LIN-12 signaling network [21]. This model was based on the
current understanding of the bidirectional coupling between
LIN-3 and LIN-12 signaling pathways (Figure S1A). To make
the model tractable, we represented multistage signaling
cascades and redundant pathways as a single reaction pathway.
This coarse-grained representation completely maintains the
regulatory logic of the LIN-3/LIN-12 network, while simplifying
its mathematical representation. Distinct from other modeling
strategies [12,13], this mathematical model formally encodes the
quantitative strength of every molecular interaction in the
regulatory network, a necessary feature to probe quantitative
diversification.
Ordinary differential equations were formulated to track the level
of two fate-encoding signals in each cell i: active MAP kinase
(MAPK) molecules (mpk�i ) and lateral signal activity (lati). These
equations are:
d mpk�i� �
dt~kz
m Indi mpkið Þ{k{m PhTð Þ mpk�i
� �
{kx1
latið Þ2
K2Mlat
z latið Þ2mpk�i� �
d latið Þdt
~kzn {k{
n latið Þ{kx2mpk�i� �
latið Þ
zkx3
mpk�iz1
2z
mpk�i{1
2
� �2
K2Mind
zmpk�iz1
2z
mpk�i{1
2
� �2
ð1Þ
Figure 7. Quantitative differences predicted to have arisen during the evolution of the Caenorhabditis genus. The likelihood that C.remanei (A) or C. briggsae (B) differ from C. elegans by higher (open columns) or lower (filled columns) values for parameters indicated on the x-axis.doi:10.1371/journal.pcbi.1000354.g007
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where ni is the number of neighbors for cell i and the other
dimensional parameters are described in the legend to Figure S1A.
In addition, each VPC is stimulated by a local amount of
inductive signal, Indi. The values for Indi were determined by
modeling diffusive transport of the soluble factor coupled with
linear degradation in the extracellular space. At steady state, the
gradient is described by:
0~DL2 Ind½ �
Lx2{kd Ind½ �, ð2Þ
whose solution is:
Ind½ � xð Þ~ IndP6:p
� �e{
ffiffiffikdD
px, ð3Þ
when we require that Ind½ � x~0ð Þ~ IndP6:p
� �. We rewrite this
solution by rescaling the spatial axis, x, in terms of the length of
P3.p-P6.p VPC field, L, as follows:
Ind½ � ~xxð Þ~ IndP6:p
� �e{
ffiffiffikdD
pL2
~xx3~ IndP6:p
� �DI
~xx3, ð4Þ
where ~xx is 0, 1, 2 and 3 for P6.p, P5/7.p, P4/8.p and P3.p,
respectively. Thus, the parameters IndP6.p and DI specify the local
level of inductive signal (Indi). A change in the value of DI alters
the steepness of the exponential gradient in inductive signal.
The dimensional variables mpk�i and lati were normalized by
their characteristic values, mpkT and latT, respectively to yield the
following nondimensional state variables:
mi~mpk�impkT
, li~lati
latT: ð5Þ
Subsequently, dimensional parameters in the model equations
were rearranged to identify the following dimensionless parameter
groups:
t~k{m PhTð Þt, I~
kzm IndP6:p
� �k{
m PhTð Þ , x~kx1
k{m PhTð Þ ,
l~kz
n
k{n latTð Þ , w~
kx3
k{n latTð Þ , h~
kx2mpkTð Þk{
n
,
kM~KMind
mpkT, kL~
KMlat
latT, c~
k{n
k{m PhTð Þ :
ð6Þ
Thus, by using non-dimensional parameters, we have reduced
the space of parameters from 13 dimensional parameters to 9
dimensionless ones. This reduction reduces the computational
load albeit this load is not prohibitive as others have analyzed
parameteric sensitivity of biomolecular networks by sweeping
across 36 dimensional parameters[37]. Using these nondimen-
sional quantities, our model equations may be rewritten as follows:
dmi
dt~ DIð Þ
~xx3I 1{mið Þ{mi{xmi
l2i
k2Lzl2
i
,
dli
dt~c l{li{hmilizw
mi{1
ni{1z miz1
niz1
� �2
k2Mz mi{1
ni{1z miz1
niz1
� �2
264
375:
ð7Þ
Framework for assigning cell fates. The timing of VPC
patterning has been studied by ablating the anchor cell (AC) at
different times during the induction process. Results from these
experiments have established that the AC (and therefore, the LIN-
3 signal that it secretes) is needed for approximately 6 hours in
order for the VPCs to commit to the 3u3u2u1u2u3u fate pattern
[26,27]. Our model calculations show that the fate-determining
signals (MAP kinase (mi) and lateral (li) signals) reach their steady-
state values within 5 hours for reference parameter values (detailed
below). Therefore, we worked under the reasonable assumption
that the steady-state values of mi and li prescribe the fate choice of
each VPC. For all simulations, the steady-state solution of the
dimensionless model equations was determined using the initial
condition that the levels of inductive and lateral signal are zero in
all cells. We note that for steady-state calculations the
dimensionless group c is eliminated from model equations (7).
The output of each simulation is the dimensionless magnitudes
of the fate-determining signals (mi, li). These are in turn recast into
the dimensional form (mpk�i , lati) from which fate assignments are
determined using the framework that we described previously
(Figure S1B) [21]. Briefly, (mpk�i , lati) in each VPC is a point in
the (mpk�, lat) fate plane. Two orthogonal thresholds, (mpk�Th,
latTh) segregate the fate plane into four quadrants. The
dimensional inductive and lateral signals in each cell are compared
against their respective threshold values, which then translate into
1u, 2u, 3u or m fate quadrants (Table S1).
Quantifying phenotypic capacityIn order to explore phenotypes that would result from
quantitative variations in network performance, we varied the
value of each dimensionless parameter, starting from its central
value and expanding in a step-wise fashion by increasing and
decreasing its value by ,3–4 fold. The central values of the
dimensionless parameters were determined as described in
Supporting Text S1. In this manner, the parameter space
hypervolume was expanded sequentially and contained 38, 58,
78, 98 and ultimately 118 points. Therefore, at its maximum size,
the parameter space contained 11 values per parameter (equally
spaced on a log scale), spanned 5–6 orders of magnitude for each
parameter (Tables S2, S3), and represented 118 parameter
combinations in total.
For each combination of 8 model parameter values, we
computed the fate pattern. Importantly, the fate of each cell i is
determined by whether the amounts of MAP kinase and lateral
signals in that cell (mpki and lati) exceed threshold levels (mpk�Th
and latTh, respectively; see Table S1). Because these threshold
values are unknown, and in fact, may be a source of variation in an
evolutionary context, we computed fate patterns across a broad
range of threshold values. Specifically, mpk�Th and latTh were
varied across the ranges 0ƒmpk�Thƒ10,000 molec=cell and
0ƒlatThƒ100,000 molec=cell, respectively. The cumulative
number of fates predicted across the 8-dimensional parameter
space for every combination of threshold values is reported in
Figure 2C.
Calculating the Parameter Space Occupancy (PSO)To quantify the PSO for each phenotype, we determined the
number of parameter points associated with each phenotype at
every combination of threshold values. This total level of
occurrence of each phenotype was divided by the total number
of parameter points to yield the fraction of parameter space
occupied by that particular phenotype. Phenotypes were binned
according to the fraction of parameter space occupied in unit log10
bins (i.e., 1 to 0.1, 0.1 to 0.01, etc). The number of distinct
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phenotypes in each bin is plotted on the y-axis in Figure 2D. The
distribution of parameter space occupancy was then fit to a log-
normal probability distribution. There are 19 phenotypes two
standard deviations below the mean (Table S4) and 34 phenotypes
two standard deviations above the mean (Table S5).
Quantifying the robustness of the phenotype subspacesto parameter variations: the Connectivity and Shape (CS)and the Mean Path Length (MPL) metrics
Each point in the 8-dimensional parameter space maps to a
phenotype (Figure 2B). We refer to the collection of points in the
parameter space that are associated with a particular phenotype as
the phenotype subspace. To quantify the CS value for each
phenotype, we distinguished between isolated, edge, and interior
points in the phenotype subspace. Isolated points are those points
for which unit jumps along both (increase and decrease) directions
of every parameter axis lead to points associated with another
phenotype. In the other extreme, there are interior points for
which unit jumps in both directions along every parameter axis reach
points that still belongs to the same phenotype. Finally, between
these possibilities are edge points: a unit jump in at least one
direction along at least one parameter axis leads to another
phenotype. To calculate the CS metric for a phenotype, we assign
each point in the phenotype subspace a score equal to the number
of neighboring points that belong to the same phenotype. This
score ranges between 0 (for isolated points) and 16 (for interior
points). We add the scores of each point in the phenotype subspace
and normalize this total by the maximum possible score for the
phenotype space, accounting for edge effects due to finite
parameter domains. This normalized score is the CS value plotted
in Figure 3C.
A complementary approach to gauge robustness is to quantify
how easy it is to drift out of the phenotype subspace by computing
the MPL of escape from the phenotype subspace. We choose
randomly a point in the subspace and then make unit jumps along
a randomly selected parameter axis and direction. We record the
number of jumps taken before exiting the phenotype. This process
is repeated until the running average number of jumps stabilizes.
We conduct 10 such drift trial reseeding the random number
generator between trials. The mean path length is the average over
these 10 trials.
Importantly, the 8-dimensional phenotype phase diagram will
be sensitive to the threshold values of MAPK (mpk�Th) and lateral
(latTh) signals. Recall that these thresholds determine how fates are
assigned (Table S1). Hence, we computed the MPL and CS
metrics across 25 different threshold combinations spanning the
following ranges:
mpk�Th[ 1000,2000,3000,4000,5000½ � molecules=cell
latTh[ 10000,20000,30000,40000,50000½ � molecules=cell
Figure 3C reports the average and standard error across these 25
threshold combinations.
Predicting the most effective molecular perturbations forrendering mutant phenotypes: the transition probability
Each phenotype, including the wild type, occupies a subspace
within the 8-dimensional parameter space (Figure 2B). This phase
diagram of phenotypes was analyzed to address the following
question: given a choice of 8 single mutations (i.e., 8 parameter
perturbations), which single-parameter change (i.e., single muta-
tion) would be most likely to promote a transition from wild-type
(W) to a mutant (M) phenotype? To address this question, we rank
ordered the parameters according to their relative transition
probabilities (Figures 4B and 4C), computed as described below.
The same transition probability metric is computed to quantify the
single-parameter differences that distinguish C. elegans from closely
related species (Figures 7A and 7B). For this analysis, ‘‘transitions’’
between parameter spaces associated with C. elegans and another
species (C. briggsae or C. remanei) were considered.
For the purpose of this discussion, let Pk denote each
dimensionless parameter where k = 1 to 8. Let i denote a point
in the W parameter space, and j denote a point in the M-
parameter space (Figure 4A). By scanning through all (i, j) pairs,
we determined the total number that differ only by a single
parameter value. These pairs represent the cases where a single-
parameter change can cause a WRM phenotype transition.
Among this total number of single-mutation pairs, we determined
the fraction of phenotype transitions that are attributable to an
increase in a particular parameter Pk. This fraction is the transition
probability of WRM phenotype transition by increasing Pk. The
same calculation was conducted for quantifying the transition
probability due to a decrease in Pk.
To determine the robustness of the transition probability to
variations in the fate-determining thresholds, we computed the
transition probability for 25 different threshold combinations
presented above. Hence, the y-axes of Figures 4B, 4C, 7A, and 7B
report the mean transition probability computed over all these 25
threshold combinations, and the error bar denotes the standard
deviation.
Predicting the phenotypes accessible throughquantitative changes in the level of inductive signal
Starting from the wild-type phenotype, we determined all the
mutant phenotypes that may be rendered solely by increasing (or
decreasing) the inductive signal. Since some mutant phenotypes
are more prevalent than others, we quantified the likelihood that
an increase (or decrease) in inductive signal would produce each
mutant (M). To quantify this likelihood of phenotype occurrence,
we first tallied the total number of ways that a change in inductive
signal (I) would abolish the wild-type (W) phenotype. Among this
total, we quantified the fraction that shifted W to a specific mutant
M upon an increase (or decrease) in I. This fraction represents the
likelihood of producing M phenotype by an increase (or decrease)
in inductive signal (I).
Phenotype assignments must be sensitive to fate-determining
threshold values of MAPK and lateral signals (Table S1). To
quantify the robustness of the likelihood of phenotype occurrence
to threshold variations, we performed the calculation for 25
different threshold combinations (as described above). The mean
of the likelihood of phenotype occurrence is reported in Figure 5A
and Figure S2A, and the error bars denote the standard deviation.
Figure 5A shows the mutant phenotypes with the greatest
likelihood of phenotype occurrence upon an increase (empty) or
decrease (filled) in inductive signal. The more complete set of
phenotypes, including the ones that occur less frequently, are
shown in Figure S2A. Similar calculations were performed to
determine the phenotype diversity due to changes in gradient
steepness. Figure S3 shows the mutant phenotypes with greatest
likelihood of phenotype occurrence upon an increase (empty) and
decrease (filled) in gradient steepness. Note the occurrence of
1u2u2u1u2u1u and 2u1u2u1u2u1u phenotypes in both Figure 5A and
Figure S3.
In addition to the likelihood of generating a particular mutant
phenotype, it is also important to gauge the amount of change in
inductive signal needed to render each mutant. Some mutant
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phenotypes may require only small changes, while others may
require substantial perturbations. Therefore, we quantified the
fold change in I needed to produce a specific mutant phenotype
(M). For every increase (or decrease) in I that produced phenotype
M, we kept track of the associated magnitude of change in I. The
geometric mean of these magnitudes was computed to give the
fold change in I. As with other calculations, we examined the
robustness of this quantity to variations in fate-determining
thresholds. The mean fold change in I across a broad range of
threshold settings is reported in Figure 5B and Figure S2B, and the
error bars represent the standard deviation.
Partitioning the wild-type subspace into species-specificregions
A key experimental observation is that changes in inductive
signal produce species-specific phenotypes [22]. Figure S4
highlights the progression of phenotypes observed in C. elegans,
C. briggsae, and C. remanei along the inductive signal axis. We
developed a computational approach to analyze how these
experimental phenotypes are arranged in our predicted phase
diagram of phenotypes with the goal of identifying the regions
within the wild-type subspace that belongs to each species.
First, we designated each phenotype with a letter code (Figure
S4), so that a string of characters or a word may be used to
represent the phenotype progression of each species. Phenotypes
that are not described in Figure S4 were designated ‘O’. For
example, following the lines for C. elegans in Figure S4, one word is
APWRD. Using this nomenclature, we identified the words that
are consistent with the fate progression observed experimentally in
C. elegans, C. briggsae, and C. remanei (Table S6).
Next, we determined the word associated with every predicted
point in the wild-type subspace. To construct the word, we varied
the value of I from its minimum to maximum while holding all
other parameter values constant. As the I-axis was traversed, we
recorded each phenotype with its character designation, thereby
yielding a 11-character word (11 characters because of the
discretization of the I-axis). The length of these words was then
condensed by eliminating adjacent repeats of a character. For
example, APPPOWOSSDD would become APOWOSD (Figure
S5). Since ‘O’ phenotypes include cases where VPCs are
designated as ‘m’ fate (a fate whose experimental equivalent
remains to be elucidated), we removed ‘O’ from the predicted
words. In the example, APOWOSD would become APWSD.
Thus, at the end of this step, every point in the wild-type
parameter is associated with a word that characterizes how the
phenotype would change when I is increased or decreased.
Finally, we compared the predicted words associated with each
point in wild-type parameter space with the experimentally
observed phenotype progressions/words of C. elegans, C. briggsae,
and C. remanei. In this manner, we identified the regions within the
wild-type parameter space associated with each species.
Supporting Information
Text S1 Rationale for the central values of dimensionless
parameters
Found at: doi:10.1371/journal.pcbi.1000354.s001 (0.32 MB PDF)
Figure S1 Model schematic of regulatory network and fate
assignments
Found at: doi:10.1371/journal.pcbi.1000354.s002 (0.43 MB PDF)
Figure S2 Extended set of phenotypes that occur upon changing
the level of inductive signal
Found at: doi:10.1371/journal.pcbi.1000354.s003 (0.38 MB PDF)
Figure S3 Phenotypic diversity caused by quantitative changes
in gradient steepness
Found at: doi:10.1371/journal.pcbi.1000354.s004 (0.09 MB PDF)
Figure S4 Letter representations of the phenotypes observed in
C. elegans, C. briggsae and C. remanei
Found at: doi:10.1371/journal.pcbi.1000354.s005 (0.29 MB PDF)
Figure S5 An illustration of our word representation for the
order of phenotypes that occurs as inductive signal is increased
Found at: doi:10.1371/journal.pcbi.1000354.s006 (0.10 MB PDF)
Table S1 Fate assignment based on threshold values
Found at: doi:10.1371/journal.pcbi.1000354.s007 (0.08 MB PDF)
Table S2 The values of dimensional parameters used to
determine the center values for the dimensionless parameters
Found at: doi:10.1371/journal.pcbi.1000354.s008 (0.18 MB PDF)
Table S3 Range of values for dimensionless model parameters
Found at: doi:10.1371/journal.pcbi.1000354.s009 (0.06 MB PDF)
Table S4 List of phenotypes with PSO values that are two
standard deviations below the mean
Found at: doi:10.1371/journal.pcbi.1000354.s010 (0.04 MB PDF)
Table S5 List of phenotypes with PSO values that are two
standard deviations above the mean
Found at: doi:10.1371/journal.pcbi.1000354.s011 (0.04 MB PDF)
Table S6 Characteristic words associated with each species
Found at: doi:10.1371/journal.pcbi.1000354.s012 (0.05 MB PDF)
Author Contributions
Conceived and designed the experiments: CAG. Performed the experi-
ments: CAG. Analyzed the data: CAG PWS ARA. Contributed reagents/
materials/analysis tools: PWS. Wrote the paper: CAG ARA.
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